Laboratory demonstrations and supporting theory Fluid Dynamics Atmospheric Sciences 505, Oceanography 511 Fall Q.! 2014 P.B. Rhines Prof. of Oceanography and Atmospheric Sciences University of! Washington These are notes in support of the lecture course offered this year by Prof. Chris Bretherton. Students came from Atmospheric Sciences, Applied Mathematics and Oceanography, and this is !a particularly important preparation for specialized courses like Geophysical Fluid Dynamics. The lab makes it possible to see essentially all the key ideas in these courses (and then some). Over about 25 years we have found it to be an effective complement to the classroom, as well as stimulating new research ideas. Despite the ever more accurate computer models of fluids in action, there are many reasons to continue experimenting with real fluids. One of these involves the basic physics (and chemistry) of water in all its phases. Water is a remarkably unusual !substance yet like Mona Lisa’s smile, it is not easy to see it afresh. I hope to continue writing down these experiments and the theoretical ideas that surround them, as we proceed into this year’s GFD series, come Winter and Spring 2015. Bundled !together with videos from the lab, it might turn into an interesting volume. I am indebted to Eric Lindahl and Bob Koon, who worked as engineers in my GFD lab for many years, and provided not only technical expertise, but some of the most creative research and effective teaching of students that I have ever seen. Peter Rhines, Seattle, 5 Dec 2014 http://www.ocean.washington.edu/gfd AS505/Ocean 511 Lab #1a: pressure, buoyancy, Archimedes Principle, hydrostatic balance ! P.Rhines notes 13x2014 !Summary o Archimedes principle and hydrostatic balance: water at rest presses on a solid object with the same force that it presses on water, if the object is removed (the ‘weight of water displaced by the object’). Weigh a solid object in air, and then weight it immersed in water: the difference in weight is the object’s volume times the fluid density…that is Archimedes’ buoyancy force. o The upward buoyancy force on solids or on the fluid itself is the result of the hydrostatic increase in pressure with depth, which is proportional to fluid density: a ‘spar buoy’ !or hydrometer tells you that fluid density. o In many geophysical fluid flows the pressure is nearly hydrostatic: a spar buoy oscillates up and down when disturbed, with frequency calculated from hydrostatic balance. Fluid in a U-shaped tube oscillates (as a ‘fluid pendulum’) when disturbed. If the U is much wider than it is tall, the frequency (2g/L)1/2 can be found by assuming the pressure to be hydrostatic. L is the length of the horizontal part of the U. Using Bernoulli’s equation, that frequency turns out to be exactly correct if L is taken to be the length of the whole fluid segment (vertical and horizontal); in many cases, fluid motions varying horizontally over distance L and varying vertically over distance H are nearly hydrostatic if (H/L)2 << 1…wide aspect ratio. Pressure is hard to ‘see’, but the elevation pattern of a water surface tells us the pressure, !when it is close to hydrostatic…as in long gravity waves. o pressure forces arise from impacts of molecules on a solid boundary or the flux of the momentum of molecules through an imaginary control surface within the fluid: for an ideal, dilute gas the equation of state follows immediately, provided we define temperature as proportional to the kinetic energy of translation of gas molecules. Polyatomic gases can store energy in vibrational and rotational modes of motion as well as their KE..roughly the same amount of energy in each degree of freedom: this leads to the great heat capacity of such gases, compared with monatomic or diatomic gases. Knowing the numbers for specific heat capacity thus tells us the average speed of air molecules: 500 m sec-1 for air at room temperature. Air is a mixture, and with the kinetic energy (i.e. the temperature) the same for each species of molecule, there is a wide range of rms speeds from the 21% of heavier, slower oxygen molecules to the 78% of lighter, faster nitrogen molecules. We measure Kelvin temperature starting at 0K which is -2730C. In the lab we used a flask connected to a glass syringe to plot the volume of air against temperature at constant pressure (atmospheric), from 00C to 400C. For an ideal gas this defines at straight line, which intersects the zero axis for density at absolute zero Kelvin! We missed the number but when done carefully it predicts -2730C quite well. o Several forms of the equation of state of air are useful : P(ρ,T), P(S,ρ), P(θ, ρ). (S = entropy). Adiabatic increase in temperature when fluid is compressed is very large: the absolute (Kelvin) temperature increases by a factor 1.32 when the fluid is compressed to 1/2 its initial volume. In the lab, smaller adiabatic warming is visible in air jets striking a boundary, or in a syringe of air that is compressed, with a thermistor (resistance thermometer) inside: the temperature rises according to γ-1 0.4 T/T0=(ρ/ρ0) = (ρ/ρ0) o Moist air is dynamically active owing to phase change of water: evaporative cooling, warming by condensation: visible with liquid crystal thermometer sheets. It takes 2.5 million Joules of energy to evaporate 1 kg of water at room temperature. (That’s 1 kilowatt heating for 42 minutes). You are burnt (scalded) by steam when it condenses on your hand, not so much by its initially hot temperature. A glass tube with water and bubbles of water vapor, evacuated to very low pressure, with the air removed, has strange properties: the ‘clinker’. Vapor bubbles (not air) appear in this near-vacuum, and disappear as pressure changes due to shaking; warming one end of the tube evaporates water, pushing the column of fluid upward: this tells us the vapor pressure of water, which increases exponentially with temperature. This becomes a ‘heat pipe’ with vapor bubbles transferring heat from the evaporating warm end to where they condense at the cool end…as in the poleward flux of moist thermal energy in the atmosphere. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Elaboration: (These are notes meant to archive our lab demonstrations for the future, providing some detail !behind the experiments, hence they are rather long). Pressure exerts forces normal to a boundary, and is the only surface-contact force in a fluid at rest (viscous forces occur due to motion in a Newtonian fluid, including forces both tangential and normal to a boundary. The ‘boundary’ can be an imagined surface within the fluid. Pressure is a scalar, its force on a fluid is -∇p per unit volume. This body force can be integrated over a volume to give the normal surface pressure force, ∯pn dA where n is an inward pointing unit vector normal to the boundary, with area element dA. {I’ll be talking about !the molecular origin of pressure a bit in this Lab #1b.} Hydrostatic balance: In a resting fluid, the vertical momentum equation becomes just 0 = -1/ρ ∂P/∂z - g with no vertical acceleration, Dw/Dt, term. Near the Earth’s surface gz is the gradient of the potential field Φ ≈ gz where z is a vertical unit vector and z is the vertical elevation. As with other potential force fields, the work done against ‘gravity’ as a particle moves around is given by the difference in the potential Φ at the two end-points of its path, regardless of details of the path. Thus it is in fact the potential energy of the particle, per unit mass. With the complexity of the Earth’s mass field, the surfaces Φ=const. (called the ‘geoid’) are complex, roughly ellipsoidal with an Equatorial radius-bulge of 21 km. This defines the local vertical and the horizon. ! ρΦ becomes our potential energy. The resting fluid is a balance between the downward ‘gravity’ force and upward buoyancy force: with pressure increasing downward, the normal pressure force on a cube of fluid is larger at its base than on its top, by just the right amount. It often simply says that the force on the !lower boundary due to pressure is just the weight of the column of fluid overhead. But not always. For example a flask with sloping sides will have a net force on the bottom due to pressure which exceeds the weight of the water, in some cases by a large factor. While seemingly unimportant, consider a planet with a very thick atmosphere, greater than the radius of the solid planet. Will the net gravity force of the atmosphere on the planet be equal to the weight of the air? No! Isaac Newton thought this through long ago. He concluded that the force on the planet is given by the weight of air in ‘soda-straw’ like cylinders above the planet, but much of the inward gravity force is balanced by the sloping sides of sectors of air bounded by radii…like an arched stone bridge: the weight of the stones is not balanced by anything but the other adjacent stones. Or like a medieval cathedral roof with flying buttresses, which may have !been outside Newton’s window. Archimedes’ Principle: is based on the intuitive idea that a solid immersed in a fluid will feel the same pressure force on its bounding faces, as did the fluid that it replaces (or ‘displaces’). This is not obvious; if the fluid interacted at the molecular level with the solid it might exert some tangential forces…it might be ‘sticky’ fluid, perhaps non-Newtonian.